After successful completion of the course, the student is expected to be able to:
Explain the connection between linear regression, parameter estimation and inverse theory.
Derive the closed form solutions to a variety of linear least squares problems.
Explain the connection between continuous models and representers.
Derive the Singular Value Decomposition, describe the properties of the natural inverse solution and implement an algorithm for solving a simple geophysical linear inverse problem.
Describe the principles of Tikhonov regularisation and critically interpret the trade-off between resolution, bias and uncertainty of the Tikhonov solution.
Explain the basic principles behind iterative methods for solving large linear systems of equations and use the conjugate gradient method to solve a simple geophysical linear inverse problem.
Use Fourier transforms to solve the deconvolution problem by water levelling regularisation.
Give an account of and apply the basic methods for solving non-linear equations.
Solve a simple geophysical non-linear inverse problem by Occam regularisation.
Explain Bayesian approaches to inverse solutions and use apriori information to solve a simple geophysical inverse problem.
Short review of mathematical tools: linear algebra, statistics, and vector algebra; linear regression and linear inverse problems; discretisation of continuous inverse problems; the Singular Value Decomposition; Tichonov regularisation; other methods of regularisation; Fourier techniques; iterative methods, including the conjugate gradient method; non-linear regression and non-linear inverse problems, including Occam’s method; Bayesian methods.
Lectures, homework, problem solving and computer solution of simple geophysical inverse problems using MATLAB.
Oral examination (7 ECTS) and compulsory part (3 ECTS).